Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics C Solid State Physics

The eigenstates of short chains of spin-1 atoms with Heisenberg and biquadratic exchange

John B. Parkinson

Published May 28, 1977• 12 citations
Publisher

Abstract

The eigenstates of chains of up to eight spin-1 atoms interacting with both Heisenberg and biquadratic exchange are calculated numerically. The ground state is found to be ferromagnetic or antiferromagnetic for most values of the exchange but can be anomalous for certain ranges. An extension of Munro's exact results is also presented, which involves an assumption about the ground state of a system with biquadratic exchange only.

DOI
10.1088/0022-3719/10/10/015

Related papers

A finite-lattice study of the spin-1 Heisenberg model with biquartic exchange

Phillip M. Duxbury, J. Oitmaa

The spin-1 Heisenberg model with bilinear and biquadratic exchange is studied by means of exact calculations on finite cells of up to ten sites, in one and two dimensions. The authors investigate the nature of the ground state for different values and signs of the exchange constants and obtain estimates of the ground-state energy. Anomalous ground states are found for some, but not all, of the cases studied. The specific heat is also calculated.

Journal of Physics C Solid State Physics•Jul 30, 1983•2 citations

Competing bilinear and biquadratic exchange couplings in spin-1 Heisenberg chains

Jenő Sólyom

A spin-1 magnetic chain with bilinear and biquadratic exchange couplings is studied to determine its phase diagram. The character of the ground state and its degeneracy is determined from an analysis of the low-lying levels of finite chains and from the instability of the uniform lattice against dimerization. Evidence for a ferromagnetic phase, two antiferromagnetic phases (with doubled and tripled periods), and a singlet phase is found.

Physical review. B, Condensed matter•Dec 1, 1987•64 citations

Study of Interacting Heisenberg Antiferromagnet Spin-1/2 and 1 Chains

Debasmita Maiti, Dayasindhu Dey, Manoranjan Kumar

Haldane conjectures the fundamental difference in the energy spectrum of the Heisenberg antiferromagnetic (HAF) of the spin S chain is that the half-integer and the integer S chain have gapless and gapped energy spectrums, respectively. The ground state (gs) of the HAF spin-1/2 and spin-1 chains have a quasi-long-range and short-range correlation, respectively. We study the effect of the exchange interaction between an HAF spin-1/2 and an HAF spin-1 chain forming a normal ladder system and its gs properties. The inter-chain exchange interaction J⊥ can be either ferromagnetic (FM) or antiferromagnetic (AFM). Using the density matrix renormalization group method, we show that in the weak AFM/FM coupling limit of J⊥, the system behaves like two decoupled chains. However, in the large AFM J⊥ limit, the whole system can be visualized as weakly coupled spin-1/2 and spin-1 pairs which behave like an effective spin-1/2 HAF chain. In the large FM J⊥ limit, coupled spin-1/2 and spin-1 pairs can form pseudo spin-3/2 and the whole system behaves like an effective spin-3/2 HAF chain. We also derive the effective model Hamiltonian in both strong FM and AFM rung exchange coupling limits.

Condensed Matter•Jan 29, 2023•1 citation

The S = 1 quantum spin chain: some exact and numerical results

D Paczek, John B. Parkinson

Abstract There has been great interest recently in quantum spin chains with S = 1. Much of this work was stimulated by Haldane’s (1983a, b) discussion of the isotropic Heisenberg model for which he predicted the existence of a disordered ground state (i.e. one in which the correlations decay exponentially) and a gap to the first excited state at k = n. We consider a system with isotropic Heisenberg and biquadratic exchange, and a magnetic field, described by the Hamiltonian.

•Mar 21, 1991•0 citations

The spin-one Heisenberg-biquadratic quantum spin chain treated by the coupled-cluster method

R. F. Bishop, John B. Parkinson, Yang Xian

The coupled-cluster method is used to obtain the ground-state energy of the isotropic Heisenberg-biquadratic quantum spin-one chain as a function of the ratio of the magnitudes of the two terms in the Hamiltonian. Two different model states are used which are expected to be valid in different regimes. In both cases we use simple approximation schemes to obtain numerical results for the ground-state energy which are compared with results of exact diagonalizations of short chains. For both cases we are able to incorporate some of the long-range correlations explicitly, using the so-called full-SUB2 approximation schemes, and this leads to evidence of phase changes at certain points. These are discussed in the light of known and conjectured phase transitions in this system.

Journal of Physics Condensed Matter•Dec 6, 1993•21 citations

Spin-1 biquadratic chain: Magnetization curve (abstract)

J. C. Bonner

The antiferromagnetic (AFM) s=1 chain with biquadratic exchange coupling has recently been the subject of considerable attention, and some controversy. Theoretical arguments by Affleck that the model has dimerized character associated with a small spectral excitation gap have not received unanimous support from numerical calculations. For finite systems the field-dependent dispersion spectra are interesting and informative concerning the nature of the model. The excitations divide into two classes having very different character, corresponding to SzT even and SzT odd. Reflection symmetry about k=π/2 is apparent if states with a given total spin ST, rather than SzT, are plotted as a function of wavevector k, consistent with the Affleck prediction of dimerized character. The high degeneracy of the eigenstates reflects the existence of new symmetry operators which (a) shift the SzT value, (b) change wave vector k to π/2−k, and (c) shift the total spin quantum ST, of the energy eigenstates. Parkinson has recently observed that finite N states with SzT even and SzT≥2 map into states of the finite N s= 1/2 XXZ model at a special value of uniaxial anisotropy. Barber and Batchelor (unpublished) have shown the existence of an analytic mapping between states of the s=1 biquadratic chain and the 2D 9-state Potts model. They further show that all states of the Parkinson s= 1/2 XXZ model map into states of the s=1 biquadratic model. Hence, they find the ground state and first excited state energies exactly, and verify the Affleck picture. Their approach falls short of establishing complete integrability, however. Numerical studies on finite chains indicate that the T=0 magnetization curve is determined by the lowest-energy states for given SzT even, which are states of the s= 1/2 XXZ chain. Hence the magnetization curve of the s=1 AFM biquadratic chain is the same as the analytically known magnetization curve of the s= 1/2 XXZ chain!

Journal of Applied Physics•May 1, 1990•0 citations
The eigenstates of chains of up to eight spin-1 atoms interacting with both Heisenberg and biquadratic exchange are calculated numerically. The ground state is found to be ferromagnetic or antiferromagnetic for most values of the exchange but can be anomalous for certain ranges. An extension of Munro's exact results is also presented, which involves an assumption about the ground state of a system with biquadratic exchange only.
Haldane conjectures the fundamental difference in the energy spectrum of the Heisenberg antiferromagnetic (HAF) of the spin S chain is that the half-integer and the integer S chain have gapless and gapped energy spectrums, respectively. The ground state (gs) of the HAF spin-1/2 and spin-1 chains have a quasi-long-range and short-range correlation, respectively. We study the effect of the exchange interaction between an HAF spin-1/2 and an HAF spin-1 chain forming a normal ladder system and its gs properties. The inter-chain exchange interaction J⊥ can be either ferromagnetic (FM) or antiferromagnetic (AFM). Using the density matrix renormalization group method, we show that in the weak AFM/FM coupling limit of J⊥, the system behaves like two decoupled chains. However, in the large AFM J⊥ limit, the whole system can be visualized as weakly coupled spin-1/2 and spin-1 pairs which behave like an effective spin-1/2 HAF chain. In the large FM J⊥ limit, coupled spin-1/2 and spin-1 pairs can form pseudo spin-3/2 and the whole system behaves like an effective spin-3/2 HAF chain. We also derive the effective model Hamiltonian in both strong FM and AFM rung exchange coupling limits.
The coupled-cluster method is used to obtain the ground-state energy of the isotropic Heisenberg-biquadratic quantum spin-one chain as a function of the ratio of the magnitudes of the two terms in the Hamiltonian. Two different model states are used which are expected to be valid in different regimes. In both cases we use simple approximation schemes to obtain numerical results for the ground-state energy which are compared with results of exact diagonalizations of short chains. For both cases we are able to incorporate some of the long-range correlations explicitly, using the so-called full-SUB2 approximation schemes, and this leads to evidence of phase changes at certain points. These are discussed in the light of known and conjectured phase transitions in this system.
The antiferromagnetic (AFM) s=1 chain with biquadratic exchange coupling has recently been the subject of considerable attention, and some controversy. Theoretical arguments by Affleck that the model has dimerized character associated with a small spectral excitation gap have not received unanimous support from numerical calculations. For finite systems the field-dependent dispersion spectra are interesting and informative concerning the nature of the model. The excitations divide into two classes having very different character, corresponding to SzT even and SzT odd. Reflection symmetry about k=π/2 is apparent if states with a given total spin ST, rather than SzT, are plotted as a function of wavevector k, consistent with the Affleck prediction of dimerized character. The high degeneracy of the eigenstates reflects the existence of new symmetry operators which (a) shift the SzT value, (b) change wave vector k to π/2−k, and (c) shift the total spin quantum ST, of the energy eigenstates. Parkinson has recently observed that finite N states with SzT even and SzT≥2 map into states of the finite N s= 1/2 XXZ model at a special value of uniaxial anisotropy. Barber and Batchelor (unpublished) have shown the existence of an analytic mapping between states of the s=1 biquadratic chain and the 2D 9-state Potts model. They further show that all states of the Parkinson s= 1/2 XXZ model map into states of the s=1 biquadratic model. Hence, they find the ground state and first excited state energies exactly, and verify the Affleck picture. Their approach falls short of establishing complete integrability, however. Numerical studies on finite chains indicate that the T=0 magnetization curve is determined by the lowest-energy states for given SzT even, which are states of the s= 1/2 XXZ chain. Hence the magnetization curve of the s=1 AFM biquadratic chain is the same as the analytically known magnetization curve of the s= 1/2 XXZ chain!